Problem 1. Solve and draw the phase diagram with the arrows of motion (i.e. plot y against y ). Also plot y against time starting from the initial condition.

y^. + 4y = 12; y(0) = 2 

y^. +10 y = 15; y (0) = 0

y^. - y = -4; y(0) = 0

y^. = 3; y(0) = 1

Problem 2. From our example in class

Q_d = a - bP

Q_s = cP

P^. = λ (Q_d - Q_s )

λ > 0

a. If both supply and demand are positively sloped, which curve should be steeper in order to have dynamic stability. Provide the intuition behind your answer. Find the analytical and qualitative solution (assume the slopes make the equilibrium stable).

b. What if both are negatively sloped? Provide the intuition behind your answer. Find the analytical and qualitative solution (assume the slopes make the equilibrium stable).

Now assume that a = 100, b = .5, c = 1, λ = 0.01 . What is the half-life of a deviation in price, i.e. how long does it take for a given deviation to be reduced by one-half (notice that this is independent of the size of the deviation itself). The same for λ = 0.5 , what is the value of λ in a market with instantaneous price adjustment.

Problem 3. Consider another example from class about a population of bacteria growing by self-division. The way we modeled it in class assumed that when an existing bacteria divided into several others the initial bacteria also remained alive, i.e. if each bacteria splits into two bacteria, r = 2, in each time interval. We begin with 1 bacteria, then we have 1+2=3, then 3+6=9 and so forth. A more realistic approach will assume that when an existing bacteria splits into two new ones, the existing bacteria dies. Write a differential equation for this alternative process.

Problem 4. Imagine a hypothetical country in which there are only three goods; two goods are produced within the country, the other is imported from abroad. One of the goods produced domestically is not purchased domestically--it is produced exclusively for export. The other domestically produced good is sold only to domestic consumers. For simplicity, we assume there is no government.

If this country were Canada, these three products might be something like:

• Restaurant meals ? produced and consumed domestically, but neither exported nor imported.
• Mountie posters ? produced domestically, but only for export.
• Coffee ? not produced domestically, but consumed domestically and thus imported.

The table below shows the prices and quantities for these three products over various years. For restaurant meals and Mountie posters, the quantities are the value added of annual production within the country. For coffee, the quantity is the amount imported annually. There are no other products either produced or imported. You will use this information to compute various measures of aggregate output (or income) and different price indexes.

It is recommended that you load these data into a spreadsheet to answer the questions below.

a) Compute nominal GDP in Canada for each year using two different methods. Explain which two of the three available approaches you are using ? the income approach, the expenditure approach, or the value-added approach.

b) Compute real GDP in Canada for each year, using 1996 as the base year (do not use the chainweighted approach ? use the old fashioned method). Now compute real GDP in Canada for each year using 2001 as the base year (same old-fashioned approach). Explain any differences in the pattern these two series take over time.

c) Explain in general why the real and nominal measures of GDP will move differently. Why is there such a difference in the movements between 2001 and 2004? What is going on in this three-year period?

d) Construct an explicit Consumer Price Index, using 1996 as the base year.

e) Now construct the implicit GDP deflator for this economy, using the measures for nominal and real GDP (2001 base year) that you constructed in Questions 1 and 2. In any given year, it is given by

Implicit GDP Deflator = (Nominal GDP/ Real GDP) × 100

f) Explain why the levels of the CPI and implicit GDP deflator sometimes move in different directions.

g) Compute the rate of inflation of the CPI for each year. (Define the rate of inflation for year t to be the percentage change in the price index from year t-1 to year t.) Do the same for the rate of inflation of the GDP deflator. Explain any differences that you observe between the two measures of inflation.

h) Compute the (nominal) dollar value of exports and imports for each year. Also compute the trade surplus (or deficit) for each year.

i) For years in which there is a trade deficit, explain how this country finances its "extra" imports. For years in which there is a trade surplus, explain what this country does with its "extra" export earnings.

Problem 5. Consider three processes that change through time; x(t) , y(t) and z(t) . The first two processes grow at the constant exponential rates g_x and g_y respectively. Calculate the growth rate of z(t) in the following cases, where k is some constant.

a) z(t) = k. x(t)/y(t)

b) z(t) = (x(t))^ky(t )

c) z(t) = (x(t))^k/y(t)^k-1

d) z(t) =  - 1/ y(t)